Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $t = \dfrac{5}{28q + 21} \div \dfrac{2q}{12q + 9} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{5}{28q + 21} \times \dfrac{12q + 9}{2q} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 5 \times (12q + 9) } { (28q + 21) \times 2q } $ $ t = \dfrac {5 \times 3(4q + 3)} {2q \times 7(4q + 3)} $ $ t = \dfrac{15(4q + 3)}{14q(4q + 3)} $ We can cancel the $4q + 3$ so long as $4q + 3 \neq 0$ Therefore $q \neq -\dfrac{3}{4}$ $t = \dfrac{15 \cancel{(4q + 3})}{14q \cancel{(4q + 3)}} = \dfrac{15}{14q} $